Trouble shooting for covariance fitting in highly correlated data

TL;DR

This paper addresses the challenge of covariance fitting failures with highly correlated data due to small eigenvalues, proposing a new eigenmode shift method to improve fitting accuracy.

Contribution

It introduces the eigenmode shift method, a novel approach to refine fitting functions without altering the covariance matrix, enhancing covariance fitting in highly correlated datasets.

Findings

Eigenmode shift method improves fit quality.

Traditional methods like diagonal approximation have limitations.

Application to $B_K$ data demonstrates effectiveness.

Abstract

We report a possible solution to the trouble that the covariance fitting fails when the data is highly correlated and the covariance matrix has small eigenvalues. As an example, we choose the data analysis of highly correlated $B_K$ data on the basis of the SU(2) staggered chiral perturbation theory. Basically, the essence of the problem is that we do not have an accurate fitting function so that we cannot fit the highly correlated and precise data. When some eigenvalues of the covariance matrix are small, even a tiny error of fitting function can produce large chi-square and spoil the fitting procedure. We have applied a number of prescriptions available in the market such as diagonal approximation and cutoff method. In addition, we present a new method, the eigenmode shift method which fine-tunes the fitting function while keeping the covariance matrix untouched.